On the Schur-Szegö composition of polynomials

Analysis
On the Schur-Szegö composition of polynomials
Vladimir Kostov a Boris Shapiro b
a Laboratoire
J.-A. Dieudonné, UMR 6621 du CNRS, Université de Nice, Parc Valrose, F-06108 Nice, cedex 2
b Mathematics Dept., Stockholm University, S-106 91 Stockholm, Sweden
Received *****; accepted after revision +++++
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Abstract
The Schur-Szegö composition of two polynomials of degree ≤ n introduces an interesting semigroup structure
on polynomial spaces and is one of the basic tools in the analytic theory of polynomials, see [5]. In the present
paper we show how it interacts with the stratification of polynomials according to the multiplicities of their zeros
and we present the induced semigroup structure on the set of all ordered partitions of n. To cite this article: V.
Kostov, B. Shapiro, C. R. Acad. Sci. Paris, Ser. I 340 (2005).
Résumé
Sur la composition de Schur-Szegö de polynômes. La composition de Schur-Szegö de deux polynômes
de degré ≤ n introduit une structure de semi-groupe dans l’espace de polynômes et est un des outils de base
dans la théorie analytique de polynômes, voir [5]. Dans cet article nous montrons comment elle intervient dans
la stratification de polynômes selon la multiplicité de leurs racines induisant ainsi une structure de semi-groupe
sur l’ensemble des partitions ordonnées d’ordre n. Pour citer cet article : V. Kostov, B. Shapiro, C. R. Acad. Sci.
Paris, Ser. I 340 (2005).
Version française abrégée
Pn
Pn
La composition de Schur-Szegö
de deux polynômes P (x) = i=0 Cni ai xi et Q(x) = i=0 Cni bi xi est
Pn
définie par P ∗Q(x) = i=0 Cni ai bi xi , voir [5]. On désigne par P oln l’espace linéaire de tous les polynômes
de x de degré au plus n. Dans la suite nous utilisons sa base monomiale standard (xn , xn−1 , . . . , 1). A tout
polynôme P ∈ P oln on associe l’opérateur TP qui est diagonal dans cette base et qui est complètement
défini par la condition : TP (1 + x)n = P (x). Il est clair que pour P (x) = Cn0 a0 + Cn1 a1 x + · · · + Cnn an xn on
a TP (xi ) = ai , i = 0, 1, . . . , n. Etant donné P comme ci-dessus on appelle la suite {ai } ls suite diagonale
Email addresses: [email protected] (Vladimir Kostov), [email protected] (Boris Shapiro).
Preprint submitted to Elsevier Science
12 mai 2006
de P . Tous deux tels opérateurs TP et TQ commutent et leur produit TP TQ correspond à la composition
de Schur-Szegö P ∗ Q. Le théorème de composition de Schur et Szegö (voir l’original dans [5] et §3.4 de
[4] ou §2 de [1]) est formulé comme suit :
Théorème 0.1 Etant donne l’image K (qui contient toutes les racines de P ) du disque unité par une
homographie, toute racine de P ∗ Q est le produit d’une racine de Q et de −γ où γ ∈ K.
Un polynôme P ∈ P oln est dit hyperbolique si toutes ses racines sont réelles. On désigne par Hypn ⊂
−
P oln l’ensemble de tous les polynômes hyperboliques et par Hyp+
n ⊂ Hypn (resp. Hypn ⊂ Hypn )
l’ensemble des polynômes hyperboliques dont toutes les racines sont > 0 (resp. < 0). On désigne par
Hu,v,w ⊂ Hypn (où u, v, w ∈ N ∪ 0, u + v + w = n) l’ensemble des polynômes hyperboliques ayant u
racines < 0, w racines > 0 et une racine 0 de multiplicité v.
Proposition 0.2 (Théorème 5.5.5 et Corollaire 5.5.10 de [4]) Si P, Q ∈ Hypn et si Q ∈ Hyp+
n
ou Q ∈ Hyp−
n , alors P ∗ Q ∈ Hypn . De plus, toutes les racines de P ∗ Q appartiennent à [−M, −m] où
M et m sont le produit maximal et minimal d’une racine de P et d’une racine de Q.
Une suite diagonale (ou un opérateur T : P oln → P oln qui est diagonal dans la base standard) est dite
une suite finie de multiplicateurs (SFM), voir [3], si elle envoie tout polynôme hyperbolique en polynôme
hyperbolique. L’ensemble Mn de toutes les SFM est un semigroupe. La caractérisation suivante des SFM
est donnée dans [2], Theorem 3.7 et dans [1], Theorem 3.1.
Théorème 0.3 Pour T ∈End(P olnR ) les deux conditions suivantes sont équivalentes :
(i) T est une SFM ;
(ii) Toutes les racines non-nulles du polynôme PT (x) =
Pn
j=0
Cnj γj xj sont de même signe.
Dans cette note on étudie la relation entre les multiplicités des racines de P , Q et P ∗ Q.
Proposition 0.4 Etant donné deux polynômes (complexes) P et Q de degré n tels que xP , xQ sont des
racines respectivement de P , Q de multiplicité mP , mQ où mP + mQ ≥ n, on a que −xP xQ est racine
de P ∗ Q de multiplicité mP + mQ − n. (Si mP + mQ = n, alors, −xP xQ n’est pas racine de P ∗ Q.)
Remarque 1 Si mP > 0, mQ > 0 et mP + mQ < n, alors −xP xQ peut être ou pas être racine de P ∗ Q.
Exemple : ((x − 1)(x − 2)(x − 3)) ∗ ((x − 1)(x − 4)(x − d)) admet −1 comme racine si et seulement si
d = 17/23.
Proposition 0.5 Pour tout P ∈ Hu,v,w et pour tout Q ∈ Hyp−
n on a P ∗ Q ∈ Hu,v,w . En particulier,
est
un
semigroupe
p.r.
à
la
composition
Schur-Szegö.
Hyp−
n
Les racines de P , Q et P ∗ Q de Proposition 0.4 (c. à d. celles de la forme −xP xQ la somme des
multiplicités de xP et de xQ étant > n) sont dites A-racines, les autres racines de P , Q et P ∗ Q sont
dites B-racines. Avec l’exception de 0 – si 0 est racine de P , on le considère comme A-racine de P et de
P ∗ Q. On associe a tout P ∈ Hypn la partition ordonnée de n dite son vecteur multiplicité V MP obtenu
comme l’ensemble ordonné des multiplicités des racines de P dans l’ordre de croissance. Pour α racine de
P ∈ Hypn on désigne par [α]− (resp. [α]+ ) le nombre total de racines de P à gauche (resp. à droit) p.r.
à α et par sign(α) le signe de α.
Théorème 0.6 Pour tout P ∈ Hypn et Q ∈ Hyp−
n le vecteur multiplicité V MP ∗Q est défini de façon
unique par Proposition 0.5 et par les conditions suivantes :
(i) Pour toute A-racine α 6= 0 de P et toute A-racine β de Q on a [−αβ]− = [α]− + [β]sign(α) .
(ii) Toutes les B-racines de P ∗ Q sont simples.
Corollaire 0.7 La composition de Schur-Szegö restreinte à Hyp−
n induit une structure de semigroupe
dans l’ensemble des partitions ordonnées n. Exemples : (2, 14, 1) ∗ (5, 6, 6) = (1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 3, 1),
(1, 14, 2) ∗ (5, 6, 4, 2) = (1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1).
2
1. Introduction
Pn
Pn
The Schur-Szegö composition
of two polynomials P (x) = i=0 Cni ai xi and Q(x) = i=0 Cni bi xi is
Pn
given by P ∗ Q(x) = i=0 Cni ai bi xi , see e.g. [5]. Let P oln denote the linear space of all polynomials in x
of degree at most n. In what follows we always use its standard monomial basis B := (xn , xn−1 , . . . , 1). To
any polynomial P ∈ P oln one can associate the operator TP which acts diagonally in B and is uniquely
determined by the condition: TP (1 + x)n = P (x). Obviously, for P (x) = Cn0 a0 + Cn1 a1 x + · · · + Cnn an xn
one has TP (xi ) = ai , i = 0, 1, . . . , n. Given P as above we refer to the sequence {ai } as to the diagonal
sequence of P . Any two such operators TP and TQ commute and their product TP TQ corresponds in the
above sense exactly to the Schur-Szegö composition P ∗ Q. The famous composition theorem of Schur
and Szegö (see original [5] and e.g. §3.4 of [4] or §2 of [1]) reads:
Theorem 1.1 Given any linear-fractional image K of the unit disk containing all the roots of P one has
that any root of P ∗ Q is the product of some root of Q by −γ where γ ∈ K.
Geometric consequences of Theorem 1.1, in particular, Proposition 1.2, can be found in § 5.5 of [4]. A
polynomial P ∈ P oln is called hyperbolic if all its roots are real. Denote by Hypn ⊂ P oln the set of all
−
hyperbolic polynomials and by Hyp+
n ⊂ Hypn (resp. Hypn ⊂ Hypn ) the set of all hyperbolic polynomials
with all positive (resp. all negative) roots. Denote by Hu,v,w ⊂ Hypn (where u, v, w ∈ N∪0, u+v +w = n)
the set of all hyperbolic polynomials with u negative and w positive roots and a v-fold zero root.
Proposition 1.2 (Theorem 5.5.5 and Corollary 5.5.10 of [4]) If P, Q ∈ Hypn and if Q ∈ Hyp+
n
or Q ∈ Hyp−
n , then P ∗ Q ∈ Hypn . Moreover, all roots of P ∗ Q lie in [−M, −m] where M is the maximal
and m is the minimal pairwise product of roots of P and Q.
A diagonal sequence, (or an operator T : P oln → P oln acting diagonally in B) is called a finite multiplier
sequence (FMS), see [3], if it sends Hypn into Hypn . The set Mn of all FMS is a semigroup. For the
following characterization of FMS see [2], Theorem 3.7 or [1], Theorem 3.1.
Theorem 1.3 For T ∈End(P olkR ) the following two conditions are equivalent:
(i) T is a finite multiplier sequence;
Pk
(ii) All different from 0 roots of the polynomial PT (x) = j=0 Ckj γj xj are of the same sign.
+
−
We get by Theorem 1.3 a linear diffeomorphism of Mn and Hypn ∪ Hypn where X means the closure
of X. In this note we study the relation between the root multiplicities of P , Q and P ∗ Q.
Proposition 1.4 Given two (complex) polynomials P and Q of degree n such that xP , xQ are roots
respectively of P , Q of multiplicity mP , mQ with µ∗ := mP + mQ − n ≥ 0, one has that −xP xQ is a root
of P ∗ Q of multiplicity µ∗ . (If µ∗ = 0, then −xP xQ is not a root of P ∗ Q.)
Remark 1 If mP > 0, mQ > 0 and µ∗ < 0, then −xP xQ might or might not be a root of P ∗ Q. Example:
((x − 1)(x − 2)(x − 3)) ∗ ((x − 1)(x − 4)(x − d)) has −1 as a root if and only if d = 17/23.
−
Proposition 1.5 For any P ∈ Hu,v,w and any Q ∈ Hyp−
n one has P ∗ Q ∈ Hu,v,w . In particular, Hypn
is a semigroup w.r.t. the Schur-Szegö composition.
The roots of P , Q and P ∗ Q involved in Proposition 1.4 (i.e. those of the form −xP xQ , the sum of the
multiplicities of xP and xQ being > n) are called A-roots, the remaining roots of P , Q, P ∗ Q are called
B-roots. With one exception – if 0 is a root of P , then it is considered as A-root of P ∗ Q. Associate to
P ∈ Hypn its multiplicity vector M VP (the ordered partition of n defined by the multiplicities of the roots
of P in the increasing order). For a root α of P ∈ Hypn denote by [α]− (resp. [α]+ ) the total number of
roots of P to the left (resp. to the right) of α and by sign(α) the sign of α.
Theorem 1.6 For any P ∈ Hypn and Q ∈ Hyp−
n the multiplicity vector M VP ∗Q is uniquely determined
by Proposition 1.5 and the following conditions:
3
(i) For any A-root α 6= 0 of P and any A-root β of Q one has [−αβ]− = [α]− + [β]sign(α) .
(ii) Every B-root of P ∗ Q is simple.
Corollary 1.7 The Schur-Szegö composition restricted to Hyp−
n induces a semigroup structure on the
set of all ordered partitions of n. Examples: (2, 14, 1) ∗ (5, 6, 6) = (1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 3, 1), (1, 14, 2) ∗
(5, 6, 4, 2) = (1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1).
2. Proofs.
Proof of Proposition 1.4: Let xP and xQ be the required roots of P and Q. It suffices to consider the case
xP = xQ = −xP xQ = −1. Indeed, xP , xQ , −xP xQ are roots of P (x), Q(x), P ∗ Q(x) of multiplicities mP ,
mQ , µ∗ if and only if 1, 1, −1 are roots of P (xP x), Q(xQ x), P ∗ Q(xP xQ x) of the same multiplicities. Set
Pn−ν j
n!
(ν)
(1) =
G(x) = xn P (1/x). Hence, 1 is an mP -fold root of G. One has G(ν) (1) = (n−ν)!
j=0 Cn−ν aj , Q
P
P
P
n−ν j
s
r
n!
s! (n−s)
r! (n−r)
q
q
(1) =
(1) =
j=0 Cn−ν bj+ν . Set Ks := n! G
q=0 Cs aq , Lr := n! Q
q=0 Cr bq+n−r .
(n−ν)!
Hence, KP
=
K
=
.
.
.
=
K
=
0
=
L
=
L
=
.
.
.
=
L
,
K
=
6
0 6= Ln−mQ .
n
n−1
n−m
n
n−1
n−mQ +1
n−mP
P +1
P
n
n
One has j=0 (−1)j Cnj Kj Ln−j = j=0 (−1)j Cnj aj bj = (P ∗ Q)(−1) (∗). (Indeed, to prove the equality
between two bilinear forms in ai , bk , it suffices to set ai0 = 1, ai = 0 for i 6= i0 , i0 = 0, 1, . . . , n. The
middle part of (∗) then equals (−1)i Cni bi , one has Kj = Cji for j ≥ i, Kj = 0 for j < i, the left side
Pn−j ν
Pn
equals j=i (−1)j Cnj Cji ν=0 Cn−j
bν+j (∗∗) and one checks directly that the coefficient before bl in (∗∗)
equals (−1)i Cni if l = i and 0 if l 6= i.) Hence, if µ∗ > 0, then −1 is a root of P ∗ Q – each product
in the left side of (∗) contains a zero factor. When µ∗ = 0, then all but one products contain such a
factor, so (P ∗ Q)(−1) 6= 0. To prove that −1 is a root of P ∗ Q of multiplicity µ∗ one has to show
Pn−λ
Pn−λ
j
j
(ν)
(−1) where
for λ < µ∗ that j=0 (−1)j Cn−λ
Kjλ Ln−λ−j = j=0 (−1)j Cn−λ
aj+λ bj+λ = (n−λ)!
n! (P ∗ Q)
P
j
q
j!
λ
n−λ (λ)
(n−λ−j)
Kj = q=0 Cj aq+λ = n! (x
P (1/x))
|x=1 .
2
Proof of Proposition 1.5: We prove it in the case v = 0, i.e for any P ∈ Hypn , P (0) 6= 0 and any
Q ∈ Hyp−
n . The general case follows by continuity. The statement is trivially true for any hyperbolic
n
P (x) of degree n and Q(x) = (1 + x)n since P ∗ Q = P . Let Q ∈ Hyp−
n . Connect Q(x) to (1 + x) by
−
t
−
some path Q (x) within Hypn . (This is possible since Hypn is contractible.) Notice that if P (0) 6= 0,
then P ∗ Qt (0) 6= 0 for the whole family since the constant term of P ∗ Q is the product of the ones of P
and Q. Therefore the number of positive and negative roots of P ∗ Q is the same as for P ∗ (1 + x)n . 2
Proof of Theorem 1.6(i): Instead of (P ∗ Q)(x) we consider Z(x) := (P ∗ Q)(−x) (to have Q
the same
s
ordering of the roots on the line in all three polynomials). Suppose that P ∈ Hs,l,n−s−l , i.e. P = ( j=1 (x+
Qn
l
aj ))x ( j=s+l+1 (x − aj )), aj > 0. Fix aj for j = 1, . . . , s and deform them continuously into 0 for j =
s + l + 1, . . . , n. The MV of the negative root
Qs sets of Z does not change. Therefore to find this MV it
suffices to find it for P replaced by P1 := ( j=1 (x + aj ))xn−s . In the same way, to find the MV of the
Qn
positive root sets of Z it suffices to find it for P replaced by P2 := xs+l ( j=s+l+1 (x − aj )). When P (x)
is changed to P (−x), then Z(x) changes to Z(−x), (this explains the presence of sign(α) in (i)) and the
description of the MV of the root sets of Z can be done by considering only polynomials of the form P2 .
Consider the case when P, Q ∈ Hyp+
n (hence, P and Q play the same role). The other three cases
±
P ∈ Hyp±
n , Q ∈ Hypn can be treated by analogy using P (−x) ∗ Q(x) = P (x) ∗ Q(−x) = (P ∗ Q)(−x).
If P = (x − a)n , then Z(x) = Q(ax), so assume that each polynomial P , Q has two distinct roots,
0 < a1 < a2 and 0 < b1 < b2 , of multiplicities m1 , m2 and n1 , n2 . If n is even and m1 = n1 = n/2, then
Z has no A-roots. Recall that by Proposition 1.4 if ai bj is an A-root, then its multiplicity is mi + nj − n.
4
Assume that (one of) the biggest of the four multiplicities m1 , m2 , n1 , n2 is among the last two.
Suppose first that this is n1 . If n1 + m1 > n, n1 + m2 > n, then the root set R(Z) of Z looks like this:
(a1 b1 , V, a2 b1 , Y ), see Propositions 1.4 and 1.2. Set ](V ) = v, ](Y ) = y. When writing b1 → 0 or b1 → b2
we mean that the roots a1 , a2 , b2 are fixed. When b1 → 0, then in the limit Z has n2 non-zero roots
which are all from Y , hence, y ≥ n2 . When b2 → b1 , then in the limit Z(x) = P (b1 x) has two roots, of
multiplicities m1 and m2 . Hence, (n1 + m1 − n) + v ≥ m1 , i.e. v ≥ n2 . But v + y = 2n2 , hence, v = y = n2 .
If n1 + m1 > n ≥ n1 + m2 , then R(Z) = (a1 b1 , V ), v = n2 + m2 . If n1 + m1 ≤ n < n1 + m2 , then
R(Z) = (U, a2 b1 , V ), u + v = n2 + m1 . When b1 → 0, then v ≥ n2 because all n2 non-zero (in the limit)
roots are in V . When b2 → b1 , then (n1 + m2 − n) + v ≤ m2 , i.e. v ≤ n2 . Hence, v = n2 , u = m1 .
Let n2 = max(m1 , m2 , n1 , n2 ). If n2 + m1 > n, n2 + m2 > n, then R(Z) = (U, a1 b2 , V, a2 b2 ). When
b1 → 0, this yields u ≥ n1 , and b1 → b2 yields v ≥ n1 . As u + v = 2n1 , one has u = v = n1 . If
n2 + m1 > n ≥ n2 + m2 , then R(Z) = (U, a1 b2 , V ). When b1 → 0, this yields u ≥ n1 , and a1 → 0 implies
v ≥ m2 . As u + v = m2 + n1 , one has u = n1 , v = m2 . If n2 + m1 ≤ n < n2 + m2 , then R(Z) = (U, a2 b2 ),
u = m1 + n1 . This proves (i) of Theorem 1.6 for P, Q ∈ Hyp+
n having each ≤ 2 distinct roots.
Further we assume that P has a single A-root a, of multiplicity m. To prove the theorem by induction
on the number of distinct positive roots in P and Q it suffices to consider the result on the MV of the
root sets of Z when a multiple root of P or Q splits into two. If this is a B-root, such a splitting deforms
continuously the B-roots in Z, its A-roots and their multiplicities don’t change, and the theorem holds.
If an A-root splits into two B-roots, then it is a root of Q. Suppose that this is bjd , and that one
has bjd−1 < bjd < bjd+1 , bjν being A-roots. Denote the multiplicities of these three roots by h1 , h2 , h3 ,
and by t1 , t2 the sums of the multiplicities of the B-roots of Q from (bjd−1 , bjd ) and (bjd , bjd+1 ). Before
the splitting of bjd the polynomial Z had three A-roots stemming from bjd−1 < bjd < bjd+1 , namely,
abjd−1 < abjd < abjd+1 , of multiplicities m + hi − n, i = 1, 2, 3, with sums of the multiplicities of the
B-roots of Z from the two intervals between them equal to t1 + n − m, t2 + n − m. After the splitting
there remain only the A-roots abjd−1 < abjd+1 , the A-root abjd splits into B-roots of total multiplicity
m + h2 − n. In Q there remain the A-roots bjd−1 < bjd+1 with total multiplicity of the B-roots between
them equal to t1 + t2 + h2 . Thus the sum of the multiplicities of the B-roots of Z from (abjd−1 , abjd+1 )
after the splitting equals t1 + t2 − 2m + 2n + m + h2 − n = t1 + t2 + h2 + n − m. Hence, (i) of Theorem 1.6
holds after the splitting. If the A-root bjd is first or last, i.e. d = 1 or d = r, then the proof is similar.
Suppose that an A-root (say, c of P , of multiplicity µ) is splitting into an A-root to the left and a
B-root to the right, of multiplicities ξ and η. Then in Z there is a splitting of an A-root cf (f is an A-root
of Q) of multiplicity µ + ν − n into an A-root of multiplicity ξ + ν − n and one or several B-roots of
total multiplicity η. Suppose that at least one of these B-roots goes to the left. Shift to the left (after the
splitting) all roots of P simultaneously while keeping the ones of Q fixed. When one has c = 0, then the
number of positive roots (counted with the multiplicities) will be greater for P than for Z (this follows
from [cf ]+ = [c]+ + [f ]+ before the splitting). This is a contradiction with Proposition 1.5. Hence, all
new B-roots of Z go to the right after the splitting and one checks directly that (i) of Theorem 1.6 holds
after the splitting. If the B-root of P goes to the left, or if c is a root of Q, then the reasoning is similar.
If an A-root c splits into two A-roots c1 (left) and c2 (right) (hence, c is a root of Q), then the above
reasoning shows that in Z an A-root cf splits into two A-roots c1 f (left, f is an A-root of P ) and c2 f
(right) and one or several B-roots between them. Indeed, one shows as above that all roots different from
c1 f (resp. c2 f ) and stemming from cf must go right (resp. left). Hence, the B-roots of Z resulting from
the splitting are between c1 f and c2 f . Denote by n0 , n1 , n2 and m0 the multiplicities of c, c1 , c2 and f
(n0 = n1 + n2 ). Hence, the multiplicities of c1 f , c2 f and the total multiplicity of the B-roots of Z between
them equal n1 + m0 − n, n2 + m0 − n and n − m0 , i.e. (i) of Theorem 1.6 holds after the splitting.
Proof of Theorem 1.6(ii): We show that non-simplicity of a B-root contradicts P ∗ Q ∈ Hypn for any
P ∈ Hypn , Q ∈ Hyp±
n , see Proposition 1.2. We first settle the basic case when either P or Q has only
5
simple zeros and then use a procedure which either decreases the multiplicity of some root of P or leads
to P ∗ Q 6∈ Hypn . The multiplicity of 0 as a root of P must decrease up to 0, not to 1.
1) Basic case. Suppose that b 6= 0 is a B-root of P ∗ Q of multiplicity µ ≥ 2. If P has distinct real
non-zero roots, then such are all polynomials from a small neighbourhood ∆ of P in P olnR . If µ = 2, then
adjusting the constant term which is non-zero by assumption one can easily choose T ∈ ∆ such that T ∗ Q
have a complex conjugate pair of zeros close to b – a contradiction. If µ > 2, then one can choose T such
that (T ∗ Q)0 has a multiple root at b and (T ∗ Q)(b) 6= 0. Hence, T ∗ Q 6∈ Hypn .
2) General case. Assume that P = (x − c)l P1 (x), l ≥ 2, P1 (c) 6= 0, c 6= 0. Set Pc (x) := P (x)/(x − c).
Consider the family of hyperbolic polynomials P δ = P + δPc , δ ∈ R (†). In this family the l-tuple root
c splits into the (l − 1)-tuple root c and an extra root c − δ which is simple unless it coincides with some
other root of P . Set Uc := Pc ∗ Q. Further considerations split into 3 subcases below:
2.i) If Uc (b) 6= 0, then we find a value of δ such that P δ ∗ Q 6∈ Hypn – a contradiction. Indeed, if µ is
even, then choosing sign(δ) one obtains that P δ ∗ Q has no real roots close to b. If µ is odd, choose its
sign so that the total multiplicity of the roots of P ∗ Q close to b is < µ (when Uc0 (b) 6= 0, then P ∗ Q
can be made monotonous close to b; when Uc0 (b) = 0 6= Uc00 (b), then choose sgn(δ) so that there is a local
minimum (maximum) of P ∗ Q close to b where P ∗ Q is positive (negative); if Uc0 (b) = Uc00 (b) = 0, then
for δ 6= 0, b is a degenerate critical point and a non-root of P ∗ Q, hence, P ∗ Q 6∈ Hypn ).
2.ii) If Uc (b) = Uc0 (b) = 0, then P δ ∗ Q still has a multiple B-root at b and lower multiplicity of c.
2.iii) Suppose that Uc (b) = 0 6= Uc0 (b). Assume first that this happens for at least two distinct roots c,
d of P of which c is multiple or c = 0. Set Pcd := P/((x − c)(x − d)). Consider the 2-parameter family of
hyperbolic polynomials P δ,ε (x) = P (x) + δPc (x) + εPd (x) + δεPcd (x) (‡). In this family the root c (and
also d when multiple) splits as in (†). Observe that Pcd = (Pc − Pd )/(c − d) (+). Hence, Pcd ∗ Q(b) = 0.
Set δ = −ε(Pd ∗ Q)0 (b)/((Pc + εPcd ) ∗ Q)0 (b). For ε 6= 0 small enough one has ((Pc + εPcd ) ∗ Q)0 (b) 6= 0.
With this choice of δ the polynomial P δ,ε ∗ Q still has a multiple root at b and lower multiplicity of c.
2.iv) To finish the argument notice that the only case to consider when one cannot perform splittings
of roots of P is when P has a single multiple or zero root c (of multiplicity ν) with Uc (b) = 0 6= Uc0 (b), and
the remaining non-zero roots di of P are all simple with (Pdi ∗ Q)(b) = (Pdi ∗ Q)0 (b) = 0. The same must
be true for Q; denote by g the root of Q of multiplicity λ > 1. But then a suitable linear combination of
P and Pdi equals (x − c)n (see (+) etc.). Hence, (x − c)n ∗ Q = Q(cx) has a multiple root at b, i.e. b = cg.
As b must be a B-root (and not an A-root) of P ∗ Q, one must have ν + λ ≤ n. If ν + λ = n, then b is a
non-root of P ∗ Q by Proposition 1.4 – a contradiction. If ν + λ < n, then a suitable linear combination
of P and Pdi equals Y := (x − h)ν (x − c)n−ν for which one has that b is a multiple root of Y ∗ Q – a
contradiction with Proposition 1.4 again. 2
References
[1] T. Craven and G. Csordas, Composition theorems, multiplier sequences and complex zero decreasing sequences, to
appear in Value Distribution Theory and Its Related Topics, ed. G. Barsegian, I. Laine and C. C. Yang, Kluver Press.
[2] T. Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math., vol. 21(4), (1977), 801-817.
[3] G. Pólya and J. Schur, Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen, J. Reine
Angewm Mat., vol. 144, (1914), 89–113.
[4] Q. I. Rahman, G. Schmeisser, Analytic theory of polynomials, London Math. Soc. Monogr. (N. S.) vol. 26, Oxford
Univ. Press, New York, NY, 2002.
[5] G. Szegö, Bemerkungen zu einem Satz von J. H. Grace über die Wurzeln algebraischer Gleichungen, Math.Z., (2) vol.
13, (1922), 28-55.
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